My first attempt is this. In $\mathbb{Z}[\zeta_p]$, $1-\zeta_p$ is prime and
$$
p = \epsilon^{-1} (1-\zeta_p)^{p-1}
$$
where $\epsilon$ is a unit. Since $p$ is an odd prime, $(p-1)/2$ is an integer and
$$
\sqrt{\epsilon p } = (1-\zeta_p)^{(p-1)/2}
$$
makes sense and belongs to $\mathbb{Z}[\zeta_p]$.

How do I deal with the $\epsilon$ under the square root? I guess the condition on the congruence class of $p$ comes from that. Is this even the right way to proceed?

Uniqueness is not clear to me either. I thought about looking at the valuation $v_p$ on $\mathbb{Q}$, extending it to two possible quadratic extensions beneath $\mathbb{Q}(\zeta_p)$, then seeing how those have to extend to common valuations on $\mathbb{Q}(\zeta_p)$, but I didn't see how to make it work.

My favorite way to prove this is to explicitly write down the quadratic Gauss sum: $$g_p = \sum_{a \in \mathbb F_p} \left( \frac{a}{p} \right) \zeta_p^{a}$$
Then you can show $g_p^2 = (-1)^{\frac{p-1}{2}} p$ by direct manipulation. This gives the result very explicitly!

Recall that the discriminant of $x^n + b$ is $(-1)^{n(n-1)/2}n^n b^{n-1}$. You can calculate this using the Vandermonde determinant and some facts about power sums. If we substitute in $ n = p$ and $b = -1$ we get the discriminant of $x^p - 1$ is $(-1)^{p(p-1)/2}p^p$. Now we already know that $\Bbb{Q}(\zeta_p)$ is a splitting field for $x^p - 1$. Now recall that $$\sqrt{(-1)^{p(p-1)/2}p^p} = (-1)^{p(p-1)/2} \prod_{i < j} (\alpha_i - \alpha_j)$$

so that the expression on the left hand must be in the splitting field $\Bbb{Q}(\zeta_p)$. It follows that $\Bbb{Q}(\sqrt{(-1)^{p(p-1)/2}p^p}) =\Bbb{Q}(p^{(p-1)/2}\sqrt{(-1)^{p(p-1)/2}p})$ is a subfield of $\Bbb{Q}(\zeta_p)$. Now it is easy to see that $$\Bbb{Q}(p^{(p-1)/2}\sqrt{(-1)^{p(p-1)/2}p}) = \Bbb{Q}(\sqrt{(-1)^{p(p-1)/2}p}).$$

Hence if $p \equiv 1 \mod 4$, $(-1)^{p(p-1)/2} =1$ so that $\Bbb{Q}(\sqrt{p})$ is a subfield of $\Bbb{Q}(\zeta_p)$ and if $p \equiv 3 \mod 4$ then $(-1)^{p(p-1)/2} =-1$ so that $\Bbb{Q}(\sqrt{-p})$ is a subfield of $\Bbb{Q}(\zeta_p).$